The formal semantics of the primitive concepts of ontology and the foundations of mathematics

One of the important questions the foundations of mathematics, dating from the late 19th century, was the question of ontological status of primitive terms. Frege in his clarifying the conceptual apparatus of mathematical thinking was opposing and the new interpretation of the axiomatic method by Hilbert. The part of this controversies was the question about the status of the primitive concepts in mathematics. The cause of polemics at the turn of the 19th and 20th centuries became the status of geometry. Frege, disputing Kant's thesis on synthetic nature of the truths of arithmetic, has left the status of synthetic a priori for the truths of geometry. The publishing of Hilbert's "Foundations of geometry" marked a whole era not only in mathematics, but also in its philosophy. Hilbert's axiomatic method was the subject of discussion between him and Frege concerning the question, can we consider the axioms as a definition of concepts. When trying to determine the objects of geometry as it is presented in an axiomatic form by Hilbert, we stumble on the implicit definitions of "point", "straight" and so on. These definitions are implicit in the sense that they do not talk directly, what is the point or what is line, but rather are the specifications of the "point" is related to "line". In other words, these concepts are irrelevant to our intuition or perception of space, and all that is required is that the terms fulfill the axioms. This is a new way to address the role of axioms as hidden or implicit definitions. The ontology of mathematical systems, such as points, lines or natural numbers can be introduced in two different ways. On the one hand, for each object, a precise definition is required. On this way insist Frege and Dedekind. On the other hand, ontology is entered through the implicit definitions which characterize the entire system. This is the point of view of Hilbert. As has already been stated, from a traditional perspective, it is difficult to see how cumbersome can define ontology axioms at all. In fact, Hilbert's axioms define the ontology of course, but in a slightly different way. Frege had demonstrated that these axioms define the concepts of higher orders. There are two positions about the sufficiency of axioms for description of mathematical field. Firstly, our use of mathematical terms are captured by axioms incompletely, so we need further explanation. But it is precisely this kind of explanation has not been accepted by Hil-bert, believing that the axioms themselves. The motive for this decision is the inaccuracy of the explanations. Indeed, Frege had rendered the explanations as belonging to the "background" of science, i.e., what is not included in proper science. Hilbert believed that mathematics is self-sufficient in what she said, without further explanation. But this means that primitive terms are not fixed in advance. In other words, the primitive Hilbert's terms lacks the substantiality. They are simple schemes that must be filled with content in the application of mathematics to the outside world. In this case, however, is not quite clear how we understand the mathematical assertion. The theoretical difficulty is following: If there is a primitive simple diagrams, they are variables, but what semantics can be assigned to variables? Hilbert did not accept explanations of the primitive terms. One of the most plausible conjecture is that he was strongly influenced by the philosophy of Kant, and the idea of organizing the material experience by mind was very close to him. In this regard, the experience of having a scheme perfectly fit the role that Hilbert had put the science.

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